If I understand the problem correctly, then Reflected_Ray is the vector that points from plant coordinate frame origin towards image point (u,v). To calculate the angle that the Reflected_Ray components make in the world coordinate frame, we transform the vector components of Reflected_Ray from plant coordinate frame (PCF) to world coordinate frame (WCF). We can then simply calculate the angle formed by the X and Z components, w.r.t to the +ve x axis (or otherwise as desired).
Transform coordinates
WCF(u', v', w') = A*PCF(u, v, 0)
where A represents the Euler angle rotation matrix product to rotate the PCF unit vectors into the WCF coordinate system. The linear shift is not important here, as we are only interested in the magnitude of the unit vectors.
Calculate projected angle
theta = arctan( WCF(w') / WCF(u') )
Apologies for the terrible presentation, I'm not sure if it's possible to use MathJax in SO answers? Also, apologies if I have misinterpreted your question...